本論文以2019年台灣國際科學展覽會的獲獎作品[Yang and Chang, 2019]為研究主體，聚焦在一個圓周上多點動態系統的最小回歸時間和極限狀態。原始文獻中對於最小回歸時間的定義不夠清楚，也僅呈現有限的數值實驗結果，且有關極限狀態的證明以及結果的推廣皆有疏漏之處。因此本論文針對上述問題點加以強化與補正，藉由更精確的數學描述，定義出此動態系統的最小回歸時間。並在重新定義座標系統後，將問題聚焦於最小回歸時間為座標為1的情況，進而提出滿足該最小回歸時間的初始位置分佈條件，並就上述結果進行數值驗證。本論文最後針對極限狀態予以更嚴謹的證明，並佐以豐富的數值結果加以驗證。

In this thesis, we study a science project by Yang and Chang (2019) in the 2019 Taiwan International Science Fair, which investigated the minimum return time and the limiting states of a multi-points dynamic system on a circle. In the original report of Yang and Chang, the definition for the minimum return time of the dynamic system was not clear enough, and only a limited numerical results were reported. In addition, the proof of limiting states was not rigorous. Therefore, we feel that it is necessary to further investigate Yang and Chang’s results, aiming to a more rigorous argument with sufficient mathematical details in order to make the description of the dynamic system and its limiting behavior mathematically complete. In particular, the minimum return time of the dynamic system is now clearly defined to be the first time for the point in the system which returns to itself at the position coordinate 1. Then, our analysis provides necessary conditions on the initial distribution of the system starting form which the system can have a point that returns to itself for the first time and it occurs at position coordinate 1. Numerical experiments were conducted to verify the above conclusions, and error analysis is provided. Finally, an attempt is made to provide a more rigorous proof for the missing part of the limiting states as well as a more complete explanation to the numerical results.

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